Velocity, Speed, and Rates of Change AP Calculus Mrs. Mongold

Definitions Instantaneous Rate of Change: the instantaneous rate of change of f with respect to x at a is the derivative Motion along a line Position Function Displacement Average Velocity

Instantaneous Velocity- tells how fast an object is moving as well as the direction of motion Velocity is the derivative of the position function

Velocity is the derivative of the position function Instantaneous Velocity- tells how fast an object is moving as well as the direction of motion Velocity is the derivative of the position function When moving forward (s is increasing) and velocity is positive When moving backward (s is decreasing) and velocity is negative

Speed

Speed- is the absolute value of velocity and tells how fast regardless of direction

Acceleration -

Acceleration – the derivative of velocity

Consider a graph of displacement (distance traveled) vs. time. Average velocity can be found by taking: time (hours) distance (miles) B A The speedometer in your car does not measure average velocity, but instantaneous velocity. (The velocity at one moment in time.)

Velocity is the first derivative of position.

Speed is the absolute value of velocity. Example: Free Fall Equation Gravitational Constants: Speed is the absolute value of velocity.

Acceleration is the derivative of velocity. example: If distance is in: Velocity would be in: Acceleration would be in:

It is important to understand the relationship between a position graph, velocity and acceleration: time distance acc neg vel pos & decreasing acc neg vel neg & decreasing acc zero vel neg & constant acc zero vel pos & constant acc pos vel neg & increasing velocity zero acc pos vel pos & increasing acc zero, velocity zero

Rates of Change: Average rate of change = Instantaneous rate of change = These definitions are true for any function. ( x does not have to represent time. )

Instantaneous rate of change of the area with respect to the radius. Example 1: For a circle: Instantaneous rate of change of the area with respect to the radius. For tree ring growth, if the change in area is constant then dr must get smaller as r gets larger.

from Economics: Marginal cost is the first derivative of the cost function, and represents an approximation of the cost of producing one more unit.

Suppose it costs: to produce x stoves. Example 13: Suppose it costs: to produce x stoves. If you are currently producing 10 stoves, the 11th stove will cost approximately: Note that this is not a great approximation – Don’t let that bother you. The actual cost is: marginal cost actual cost

Note that this is not a great approximation – Don’t let that bother you. Marginal cost is a linear approximation of a curved function. For large values it gives a good approximation of the cost of producing the next item.

Things to Remember Velocity is the first derivative (how fast an object is moving as well as the direction of motion) Speed is the absolute value of the first derivative (tells how fast regardless of direction) Acceleration is the second derivative

Examples Find the rate of change of the area of a circle with respect to its radius A=πr2 Evaluate the rate of change at r = 5 and r = 10 What units would be appropriate if r is in inches

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Examples  

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Examples Find the rate of change of volume of a cube with respect to s V=s3 Evaluate rate of change at s=1 , s=3, and s=5 What would units be if measured in inches

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Example Suppose a ball is dropped from the upper observation deck, 450 m above ground. What is the velocity of the ball after 5 seconds?

Example Suppose a ball is dropped from the upper observation deck, 450 m above ground. What is the velocity of the ball after 5 seconds?  

Example Suppose a ball is dropped from the upper observation deck, 450 m above ground. What is the velocity of the ball after 5 seconds?    

Example Suppose a ball is dropped from the upper observation deck, 450 m above ground. What is the velocity of the ball after 5 seconds?      

Example Suppose a ball is dropped from the upper observation deck, 450 m above ground. What is the velocity of the ball after 5 seconds?        

Example Suppose a ball is dropped from the upper observation deck, 450 m above ground. What is the velocity of the ball after 5 seconds?          

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How high does it go?

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How high does it go? Highest when velocity=0

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How high does it go? Highest when velocity=0 v = 160 – 32t

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How high does it go? Highest when velocity=0 v = 160 – 32t 0 = 160 – 32t

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How high does it go? Highest when velocity=0 v = 160 – 32t 0 = 160 – 32t 32t=160

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How high does it go? Highest when velocity=0 v = 160 – 32t 0 = 160 – 32t 32t=160 t = 5 Max height at t = 5 so max position 160(5)-16(25)

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How high does it go? Highest when velocity=0 v = 160 – 32t 0 = 160 – 32t 32t=160 t = 5 Max height at t = 5 so max position 160(5)-16(25) The object reaches a height of 400 ft.

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? 160t – 16t2 = 256

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? 160t – 16t2 = 256 – 16t2+160t - 256 = 0

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? 160t – 16t2 = 256 – 16t2+160t - 256 = 0 16(t2 -10t +16)=0

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? 160t – 16t2 = 256 – 16t2+160t - 256 = 0 16(t2 -10t +16)=0 (t - 2)(t - 8)=0

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? 160t – 16t2 = 256 – 16t2+160t - 256 = 0 16(t2 -10t +16)=0 (t - 2)(t - 8)=0 t = 2 and t = 8

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. What is the velocity and speed of the rock when it is 256 ft above the ground on the way up? On the way down? 160t – 16t2 = 256 – 16t2+160t - 256 = 0 16(t2 -10t +16)=0 (t - 2)(t - 8)=0 t = 2 and t = 8 V(2)= 160 – 32(2) = 96 ft/sec V(8)= 160 - 32(8) = -96 ft/sec Speed at both instances are 96

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Example Note the acceleration is always downward…when the rock is   Note the acceleration is always downward…when the rock is Rising, it is slowing down and when it is falling it is speeding up.

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How long does it take for the rock to return to the ground?

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How long does it take for the rock to return to the ground? Hits the ground when position = 0

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How long does it take for the rock to return to the ground? Hits the ground when position = 0 160t - 16t2 = 0

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How long does it take for the rock to return to the ground? Hits the ground when position = 0 160t - 16t2 = 0 16t2+ - 160t = 0

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How long does it take for the rock to return to the ground? Hits the ground when position = 0 160t - 16t2 = 0 16t2 - 160t = 0 16t(t – 10) = 0

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How long does it take for the rock to return to the ground? Hits the ground when position = 0 160t - 16t2 = 0 16t2 - 160t = 0 16t(t – 10) = 0 t = 0 and 10

Example A dynamite blast propels a heavy rock straight up with a launch velocity of 160 ft/sec. It reaches a height of s = 160t – 16t2 ft. after t seconds. How long does it take for the rock to return to the ground? Hits the ground when position = 0 160t - 16t2 = 0 16t2 - 160t = 0 16t(t – 10) = 0 t = 0 and 10 The rock returns to the ground after 10 seconds

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